A055396 -id:A055396 - cp's OEIS frontend

A250469 a(1) = 1; and for n > 1, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)+1, where A055396(n) is the index of smallest prime dividing n.

1, 3, 5, 9, 7, 15, 11, 21, 25, 27, 13, 33, 17, 39, 35, 45, 19, 51, 23, 57, 55, 63, 29, 69, 49, 75, 65, 81, 31, 87, 37, 93, 85, 99, 77, 105, 41, 111, 95, 117, 43, 123, 47, 129, 115, 135, 53, 141, 121, 147, 125, 153, 59, 159, 91, 165, 145, 171, 61, 177, 67, 183, 155, 189, 119, 195, 71, 201, 175, 207, 73, 213, 79, 219, 185, 225, 143, 231, 83, 237, 205, 243, 89, 249, 133, 255

Offset: 1

Antti Karttunen, Dec 06 2014

nonn

Permutation of odd numbers.
For n >= 2, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)+1. In other words, a(n) tells which number is located immediately below n in the sieve of Eratosthenes (see A083140, A083221) in the same column of the sieve that contains n.
A250471(n) = (a(n)+1)/2 is a permutation of natural numbers.
Coincides with A003961 in all terms which are primes. - M. F. Hasler, Sep 17 2016. Note: primes are a proper subset of A280693 which gives all n such that a(n) = A003961(n). - Antti Karttunen, Mar 08 2017

Antti Karttunen, Table of n, a(n) for n = 1..5002

Cf. A000040, A003961, A016945, A046523, A055396, A078898, A083140, A083221, A084967, A249744, A249810, A249820, A249817, A249818, A250471, A266645, A280692, A280693, A283465.

Cf. A250470, A268674 (left inverses, the latter is simpler).

a[1] = 1; a[n_] := If[PrimeQ[n], NextPrime[n], m1 = p1 = FactorInteger[n][[ 1, 1]]; For[k1 = 1, m1 <= n, m1 += p1; If[m1 == n, Break[]]; If[ FactorInteger[m1][[1, 1]] == p1, k1++]]; m2 = p2 = NextPrime[p1]; For[k2 = 1, True, m2 += p2, If[FactorInteger[m2][[1, 1]] == p2, k2++]; If[k1+2 == k2, Return[m2]]]]; Array[a, 100] (* Jean-François Alcover, Mar 08 2016 *)
g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[Lookup[s, g[First@ #2] + 1][[#1]] - Boole[First@ #2 == 1] &, #] &@ Map[Position[Lookup[s, g@#], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 08 2017, Version 10 *)

a(1) = 1, a(n) = A083221(A055396(n)+1, A078898(n)).
a(n) = A249817(A003961(A249818(n))).
Other identities. For all n >= 1:
A250470(a(n)) = A268674(a(n)) = n. [A250470 and A268674 provide left inverses for this function.]
a(2n) = A016945(n-1). [Maps even numbers to the numbers of form 6n+3, in monotone order.]
a(A016945(n-1)) = A084967(n). [Which themselves are mapped to the terms of A084967, etc. Cf. the Example section of A083140.]
a(A000040(n)) = A000040(n+1). [Each prime is mapped to the next prime.]
For all n >= 2, A055396(a(n)) = A055396(n)+1. [A more general rule.]
A046523(a(n)) = A283465(n). - Antti Karttunen, Mar 08 2017

A250246 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n), a(A078898(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 45, 34, 35, 36, 37, 38, 33, 40, 41, 54, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 42, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 90, 67, 68, 135, 70, 71, 72, 73, 74, 51, 76, 77, 66, 79, 80, 99, 82, 83

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences that are permutations of the natural numbers

Inverse: A250245.
Other similar permutations: A250243, A250248, A250250, A163511, A252756.
Cf. A003961, A005843, A020639, A055396, A078898, A246278, A250470, A268674, A278524, A302042, A302046.
Differs from the "vanilla version" A249818 for the first time at n=42, where a(42) = 54, while A249818(42) = 42.
Differs from A250250 for the first time at n=73, where a(73) = 73, while A250250(73) = 103.

up_to = 16384;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A055396(n) = if(1==n,0,primepi(A020639(n)));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A250246(n) = if(1==n,n,my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k))); \\ Antti Karttunen, Apr 01 2018
(Scheme, with memoizing-macro definec from Antti Karttunen's IntSeq-library, three alternative definitions)
(definec (A250246 n) (cond ((<= n 1) n) (else (A246278bi (A055396 n) (A250246 (A078898 n)))))) ;; Code for A246278bi given in A246278
(definec (A250246 n) (cond ((<= n 1) n) ((even? n) (* 2 (A250246 (/ n 2)))) (else (A003961 (A250246 (A250470 n))))))
(define (A250246 n) (A163511 (A252756 n)))

a(1) = 1, a(n) = A246278(A055396(n), a(A078898(n))).
a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A003961(a(A250470(2n+1))). - Antti Karttunen, Jan 18 2015  - Instead of A250470, one may use A268674 in above formula. - Antti Karttunen, Apr 01 2018
As a composition of related permutations:
a(n) = A163511(A252756(n)).
Other identities. For all n >= 1:
a(n) = a(2n)/2. [The even bisection halved gives the sequence back.]
A020639(a(n)) = A020639(n) and A055396(a(n)) = A055396(n). [Preserves the smallest prime factor of n].
A001221(a(n)) = A302041(n).
A001222(a(n)) = A253557(n).
A008683(a(n)) = A302050(n).
A000005(a(n)) = A302051(n)
A010052(a(n)) = A302052(n), for n >= 1.
A056239(a(n)) = A302039(n).

A249818 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n),A078898(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 45, 34, 35, 36, 37, 38, 33, 40, 41, 42, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 54, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 66, 67, 68, 135, 70, 71, 72, 73, 74, 51, 76, 77, 78, 79, 80, 99, 82, 83, 84, 175, 86, 105

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

a(n) tells which number in square array A246278 is at the same position where n is in array A083221, the sieve of Eratosthenes. As both arrays have even numbers as their topmost row and primes as their leftmost column, both sequences are among the fixed points of this permutation.

Equally: a(n) tells which number in array A246279 is at the same position where n is in the array A083140, as they are the transposes of above two arrays.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A249817.
There are three different "deep" versions of this permutation, recursing on values of A055396(n) and/or A078898(n), namely: A250246, A250248 and A250250.
Other similar or related permutations: A249816.
Cf. A000040, A005843, A020639, A055396, A078898, A083140, A083221, A246277, A246278, A246279, A249822.
Differs from its inverse A249817 for the first time at n=33, where a(33) = 45, while A249817(33) = 39.

lim = 87; a003961[p_?PrimeQ] := a003961[p] = Prime[PrimePi@ p + 1]; a003961[1] = 1; a003961[n_] :=  a003961[n] = Times @@ (a003961[First@ #]^Last@ # &) /@ FactorInteger@ n; a055396[n_] := PrimePi[FactorInteger[n][[1, 1]]]; a078898 = Block[{nn = 90, spfs}, spfs = Table[FactorInteger[n][[1, 1]], {n, nn}]; Table[Count[Take[spfs, i], spfs[[i]]], {i, nn}]]; a246278 = NestList[Map[a003961, #] &, Table[2 k, {k, lim}], lim]; Table[a246278[[a055396@ n, a078898[[n]]]], {n, 2, lim}]
(* Michael De Vlieger, Jan 04 2016, after Harvey P. Dale at A055396 and A078898 *)

a(1) = 1, a(n) = A246278(A055396(n), A078898(n)).
a(1) = 1, a(n) = A246278(A055396(n), A249822(A055396(n), A246277(n))).
As a composition of other permutations:
a(1) = 1, and for n > 1, a(n) = 1 + A249816(n-1).
Other identities. For all n >= 1:
a(A005843(n)) = A005843(n) and a(A000040(n)) = A000040(n). [Fixes even numbers and primes, among other numbers. Cf. comments above].
A020639(a(n)) = A020639(n) and A055396(a(n)) = A055396(n). [Preserves the smallest prime factor of n].

A268674 a(1) = 1, after which, for odd numbers: a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)-1, and for even numbers: a(n) = a(A000265(n)).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 13, 4, 17, 3, 8, 7, 19, 2, 9, 11, 10, 5, 23, 6, 29, 1, 12, 13, 15, 4, 31, 17, 14, 3, 37, 8, 41, 7, 16, 19, 43, 2, 25, 9, 18, 11, 47, 10, 21, 5, 20, 23, 53, 6, 59, 29, 22, 1, 27, 12, 61, 13, 24, 15, 67, 4, 71, 31, 26, 17, 35, 14, 73, 3, 28, 37, 79, 8, 33, 41, 30, 7

Offset: 1

Antti Karttunen, Feb 11 2016

nonn

For odd numbers n > 1, a(n) tells which term is on the immediately preceding row of A083221, in the same column where n itself is.

The sequence offers a left inverse for A250469 that is slightly easier to compute than A250470.

Antti Karttunen, Table of n, a(n) for n = 1..32769
Index entries for sequences generated by sieves

Left inverse of A250469.
Cf. A000040, A000265, A001248, A055396, A078898, A083221.
Cf. also A064989.
Differs from A250470 for the first time at n=42, where a(42)=8, while A250470(42) = 10.

(* b = A250469 *) b[1] = 1; b[n_] := If[PrimeQ[n], NextPrime[n], m1 = p1 = FactorInteger[n][[1, 1]]; For[ k1 = 1, m1 <= n, m1 += p1; If[m1 == n, Break[]]; If[ FactorInteger[m1][[1, 1]] == p1, k1++]]; m2 = p2 = NextPrime[p1]; For[k2 = 1, True, m2 += p2, If[ FactorInteger[m2][[1, 1]] == p2, k2++]; If[k1 + 2 == k2, Return[m2]]]];
a[1] = a[2] = 1; a[n_?EvenQ] := a[n] = a[n/2]; a[n_] := a[n] = For[k = 1, True, k++, If[b[k] == n, Return[k]]];
Array[a, 100] (* Jean-François Alcover, Mar 14 2016 *)

a(1) = 1, after which, a(n) = a(A000265(n)) if n is even, otherwise for odd n, a(n) = A083221(A055396(n)-1, A078898(n)).
Other identities. For all n >= 1:
a(A250469(n)) = n. [This works as a left inverse for sequence A250469.]
a(2n) = a(n). [The even bisection gives the whole sequence back.]
a(2n-1) = A250470(2n-1). [Matches with A250470 on odd numbers.]
a(A000040(n+1)) = A000040(n). [Maps each odd prime to the preceding prime.]
a(A001248(n+1)) = A001248(n). [Maps each square of an odd prime to the square of the preceding prime.]

A249817 Permutation of natural numbers: a(1) = 1, a(n) = A083221(A055396(n),A246277(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 39, 34, 35, 36, 37, 38, 63, 40, 41, 42, 43, 44, 33, 46, 47, 48, 49, 50, 75, 52, 53, 54, 65, 56, 99, 58, 59, 60, 61, 62, 57, 64, 95, 66, 67, 68, 111, 70, 71, 72, 73, 74, 51, 76, 77, 78, 79, 80, 45, 82, 83, 84, 155, 86, 135

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

a(n) tells which number in square array A083221 (the sieve of Eratosthenes) is at the same position where n is in array A246278. As both arrays have even numbers as their topmost row and primes as their leftmost column, both sequences are among the fixed points of this permutation.

Equally: a(n) tells which number in array A083140 is at the same position where n is in the array A246279, as they are the transposes of above two arrays.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A249818.
There are three different "deep" versions of this permutation, recursing on values of A055396(n) and/or A246277(n), namely: A250245, A250247 and A250249.
Other similar or related permutations: A249815.
Cf. A000040, A005843, A020639, A055396, A078898, A083140, A083221, A246277, A246278, A246279, A249821.
Differs from its inverse A249818 for the first time at n=33, where a(33) = 39, while A249818(33) = 45.

lim = 87; a083221 = Table[Take[Prime[n] Select[Range[Ceiling[lim/2]^2], GCD[# Prime@ n, Product[Prime@ i, {i, 1, n - 1}]] == 1 &], Ceiling[lim/2]], {n, Ceiling[lim/2]}]; a055396[n_] PrimePi[FactorInteger[n][[1, 1]]]; a246277[n_] := Which[n == 1, 0, EvenQ@ n, n/2, True, a246277[Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ Transpose@ FactorInteger@ n, Last@ Transpose@ FactorInteger@ n]]]; Table[a083221[[a055396@ n, a246277@ n]], {n, 2, lim}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A055396 and Yasutoshi Kohmoto at A083140 *)

(define (A249817 n) (if (= 1 n) n (A083221bi (A055396 n) (A246277 n)))) ;; Code for A083221bi given in A083221
;; Alternative version:
(define (A249817 n) (if (= 1 n) n (A083221bi (A055396 n) (A249821bi (A055396 n) (A078898 n))))) ;; Code for A249821bi given in A249821.

a(1) = 1, a(n) = A083221(A055396(n), A246277(n)).
a(1) = 1, a(n) = A083221(A055396(n), A249821(A055396(n), A078898(n))).
As a composition of other permutations:
a(1) = 1, and for n > 1, a(n) = 1 + A249815(n-1).
Other identities. For all n >= 1:
a(A005843(n)) = A005843(n) and a(A000040(n)) = A000040(n). [Fixes even numbers and primes, among other numbers. Cf. comments above].
A020639(a(n)) = A020639(n) and A055396(a(n)) = A055396(n). [Preserves the smallest prime factor of n].

A250245 Permutation of natural numbers: a(1) = 1, a(n) = A083221(A055396(n),a(A246277(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 39, 34, 35, 36, 37, 38, 63, 40, 41, 54, 43, 44, 33, 46, 47, 48, 49, 50, 75, 52, 53, 42, 65, 56, 99, 58, 59, 60, 61, 62, 57, 64, 95, 78, 67, 68, 111, 70, 71, 72, 73, 74, 51, 76, 77, 126, 79, 80, 45, 82

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

The first 7-cycle occurs at: (33 39 63 57 99 81 45) which is mirrored by the cycle (66 78 126 114 198 162 90) with double-size terms.
The cycle which contains 55 as its smallest term, goes as: 55, 65, 95, 185, 425, 325, 205, 455, 395, 1055, 2945, 6035, 30845, ...
while to the other direction (A250246) it goes as: 55, 125, 245, 115, 625, 8575, 40375, ...
The cycle which contains 69 as its smallest term, goes as: 69, 111, 183, 351, 261, 273, 387, 489, 939, 1863, 909, 1161, 981, 1281, 4167, ...
while to the other direction (A250246) it goes as: 69, 135, 87, 105, 225, 207, 231, 195, 525, 1053, 3159, 24909, ...

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A250246.
Other similar permutations: A250244, A250247, A250249, A243071, A252755.
Cf. A005843, A020639, A055396, A064989, A083221, A246277, A250469.
Differs from the "vanilla version" A249817 for the first time at n=42, where a(42) = 54, while A249817(42) = 42.
Differs from A250246 for the first time at n = 33, where a(33) = 39, while A250246(33) = 45.
Differs from A250249 for the first time at n=73, where a(73) = 73, while A250249(73) = 103.

a(1) = 1, a(n) = A083221(A055396(n), a(A246277(n))).
a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A250469(a(A064989(2n+1))). - Antti Karttunen, Jan 18 2015
As a composition of related permutations:
a(n) = A252755(A243071(n)).
Other identities. For all n >= 1:
a(n) = a(2n)/2. [The even bisection halved gives the sequence back.]
A020639(a(n)) = A020639(n) and A055396(a(n)) = A055396(n). [Preserves the smallest prime factor of n].

A246675 Permutation of natural numbers: a(n) = A000079(A055396(n+1)-1) * ((2*A246277(n+1))-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 13, 10, 15, 64, 17, 128, 19, 18, 21, 256, 23, 12, 25, 14, 27, 512, 29, 1024, 31, 26, 33, 20, 35, 2048, 37, 42, 39, 4096, 41, 8192, 43, 22, 45, 16384, 47, 24, 49, 50, 51, 32768, 53, 36, 55, 66, 57, 65536, 59, 131072, 61, 38, 63, 52, 65, 262144, 67, 74, 69

Offset: 1

Antti Karttunen, Sep 01 2014

nonn

Consider the square array A246278, and also A246275 which is obtained from the former when one is subtracted from each term.
In A246278 the even numbers occur at the top row, and all the rows below that contain only odd numbers, those subsequent terms in each column having been obtained by shifting all primes present in the prime factorization of number immediately above to one larger indices with A003961.
To compute a(n): we do the same process in reverse, by shifting primes in the prime factorization of n+1 step by step to smaller primes, until after k >= 0 such shifts with A064989, the result is even, with the smallest prime present being 2.
We subtract one from this even number and shift the binary expansion of the resulting odd number k positions left (i.e. multiply it with 2^k), which will be the result of a(n).
In the essence, a(n) tells which number in the array A135764 is at the same position where n is in the array A246275. As the topmost row in both arrays is A005408 (odd numbers), they are fixed, i.e., a(2n+1) = 2n+1 for all n.
A055396(n+1) tells on which row of A246275 n is, which is equal to the row of A246278 on which n+1 is.
A246277(n+1) tells in which column of A246275 n is, which is equal to the column of A246278 in which n+1 is.

			Consider 54 = 55-1. To find 55's position in array A246278, we start shifting its prime factorization 55 = 5 * 11 = p_3 * p_5, step by step: p_2 * p_4 (= 3 * 7 = 21), until we get an even number: p_1 * p_3 = 2*5 = 10.
This tells us that 55 is on row 3 and column 5 (= 10/2) of array A246278, thus 54 occurs in the same position at array A246275. In array A135764 the same position contains number (2^(3-1)) * (10-1) = 4*9 = 36, thus a(54) = 36.

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences that are permutations of the natural numbers

Inverse: A246676.
More recursed variants: A246677, A246683.
Even bisection halved: A246679.
Other related permutations: A054582, A135764, A246274, A246275, A246276.
Cf. A000079, A003961, A005408, A055396, A064989, A246277, A246278.
a(n) differs from A156552(n+1) for the first time at n=13, where a(13) = 14, while A156552(14) = 17.

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A246675(n) = { my(k=0); n++; while((n%2), n = A064989(n); k++); n--; while(k>0, n = 2*n; k--); n; };
for(n=1, 2048, write("b246675.txt", n, " ", A246675(n)));

(define (A246675 n) (* (A000079 (- (A055396 (+ 1 n)) 1)) (-1+ (* 2 (A246277 (+ 1 n))))))

a(n) = A000079(A055396(n+1)-1)  * ((2*A246277(n+1))-1).
As a composition of related permutations:
a(n) = A135764(A246276(n)).
a(n) = A054582(A246274(n)-1).
Other identities. For all n >= 0:
a(A005408(n)) = A005408(n). [Fixes the odd numbers.]

A286164 Compound filter: a(n) = T(A055396(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

0, 2, 5, 7, 9, 16, 14, 29, 12, 16, 20, 67, 27, 16, 23, 121, 35, 67, 44, 67, 23, 16, 54, 277, 18, 16, 38, 67, 65, 436, 77, 497, 23, 16, 31, 631, 90, 16, 23, 277, 104, 436, 119, 67, 80, 16, 135, 1129, 25, 67, 23, 67, 152, 277, 31, 277, 23, 16, 170, 1771, 189, 16, 80, 2017, 31, 436, 209, 67, 23, 436, 230, 2557, 252, 16, 80, 67, 40, 436, 275, 1129, 138, 16, 299

Offset: 1

Antti Karttunen, May 04 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
MathWorld, Pairing Function

Cf. A000027, A046523, A055396, A286160, A286161, A286162, A286163.

A001511(n) = (1+valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A055396(n) = if(n==1, 0, primepi(factor(n)[1, 1])); \\ This function from Charles R Greathouse IV, Apr 23 2015
A286164(n) = (2 + ((A055396(n)+A046523(n))^2) - A055396(n) - 3*A046523(n))/2;
for(n=1, 10000, write("b286164.txt", n, " ", A286164(n)));

from sympy import primepi, isprime, primefactors, factorint
def a049084(n): return primepi(n)*(1*isprime(n))
def a055396(n): return 0 if n==1 else a049084(min(primefactors(n)))
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a(n): return T(a055396(n), a046523(n)) # Indranil Ghosh, May 05 2017

(define (A286164 n) (* (/ 1 2) (+ (expt (+ (A055396 n) (A046523 n)) 2) (- (A055396 n)) (- (* 3 (A046523 n))) 2)))

a(n) = (1/2)*(2 + ((A055396(n)+A046523(n))^2) - A055396(n) - 3*A046523(n)).

A250249 Permutation of natural numbers: a(1) = 1, a(n) = A083221(a(A055396(n)), a(A246277(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 39, 34, 35, 36, 37, 38, 63, 40, 41, 54, 43, 44, 33, 46, 47, 48, 49, 50, 75, 52, 53, 42, 65, 56, 99, 58, 59, 60, 61, 62, 57, 64, 95, 78, 67, 68, 111, 70, 71, 72, 103, 74, 51, 76, 77, 126, 79, 80, 45, 82

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

This is a "doubly-recursed" version of A249817.
For primes p_n, a(p_n) = p_{a(n)}.
The first 7-cycle occurs at: (33 39 63 57 99 81 45), which is mirrored by the cycle (66 78 126 114 198 162 90) with terms double the size and also by the cycle (137 167 307 269 523 419 197), consisting of primes (p_33, p_39, p_63, ...).

			For n = 42 = 2*3*7, we see that it occurs as the 21st term on the top row of A246278 (A055396(42) = 1 and A246277(42) = 21), recursing on both yields a(1) = 1, a(21) = 27, thus we find A083221(1,27), the 27th term on A083221's topmost row (also A005843) which is 54, thus a(42) = 54.
Examples for cases where n is a prime:
a(3709) = a(p_518) = p_{a(518)} = A000040(1162) = 9397.
a(3719) = a(p_519) = p_{a(519)} = A000040(1839) = 15767.

Antti Karttunen, Table of n, a(n) for n = 1..3718
Index entries for sequences that are permutations of the natural numbers

Inverse: A250250.
Fixed points: A250251, their complement: A249729.
Cf. A000035, A000040, A055396, A078442, A083221, A049076.
Differs from A250250 for the first time at n=33, where a(33) = 39, while A250250(33) = 45.
Differs from the "vanilla version" A249817 for the first time at n=42, where a(42) = 54, while A249817(42) = 42.

a(1) = 1, a(n) = A083221(a(A055396(n)), a(A246277(n))).
Other identities. For all n >= 1:
a(2n) = 2*a(n), or equally, a(n) = a(2n)/2. [The even bisection halved gives the sequence back].
a(p_n) = p_{a(n)}, or equally, a(n) = A049084(a(A000040(n))). [Restriction to primes induces the same sequence].
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves the "order of primeness of n"].
A000035(n) = A000035(a(n)). [Preserves the parity].

A250250 Permutation of natural numbers: a(1) = 1, a(n) = A246278(a(A055396(n)),a(A078898(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 45, 34, 35, 36, 37, 38, 33, 40, 41, 54, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 42, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 90, 67, 68, 135, 70, 71, 72, 103, 74, 51, 76, 77, 66, 79, 80, 99, 82, 83

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

This is a "doubly-recursed" version of A249818.

Antti Karttunen, Table of n, a(n) for n = 1..3432
Index entries for sequences that are permutations of the natural numbers

Inverse: A250249.
Fixed points: A250251, their complement: A249729.
Cf. A000035, A000040, A055396, A078442, A078898, A049076, A246278.
See also other (somewhat) similar permutations: A245821, A057505.
Differs from the "vanilla version" A249818 for the first time at n=42, where a(42) = 54, while A249818(42) = 42.

a(1) = 1, a(n) = A246278(a(A055396(n)), a(A078898(n))).
Other identities. For all n >= 1:
a(2n) = 2*a(n), or equally, a(n) = a(2n)/2. [The even bisection halved gives the sequence back].
a(p_n) = p_{a(n)}, or equally, a(n) = A049084(a(A000040(n))). [Restriction to primes induces the same sequence].
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves the "order of primeness of n"].
A000035(n) = A000035(a(n)). [Preserves the parity].

cp's OEIS Frontend

A250469 a(1) = 1; and for n > 1, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)+1, where A055396(n) is the index of smallest prime dividing n.

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A250246 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n), a(A078898(n))).

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A249818 Permutation of natural numbers: a(1) = 1, a(n) = A246278(A055396(n),A078898(n)).

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A268674 a(1) = 1, after which, for odd numbers: a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)-1, and for even numbers: a(n) = a(A000265(n)).

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A249817 Permutation of natural numbers: a(1) = 1, a(n) = A083221(A055396(n),A246277(n)).

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A250245 Permutation of natural numbers: a(1) = 1, a(n) = A083221(A055396(n),a(A246277(n))).

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A246675 Permutation of natural numbers: a(n) = A000079(A055396(n+1)-1) * ((2*A246277(n+1))-1).

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A286164 Compound filter: a(n) = T(A055396(n), A046523(n)), where T(n,k) is sequence A000027 used as a pairing function.

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